Dynamics of gas bubbles in liquids

Abstract

The study of bubble formation in liquids is of great importance because of its application in the design of many industrial bubble contactors like distillation columns, bubble cap columns, absorption towers, reactors, extraction apparatus, etc. The main object of the present investigation is to make an attempt to develop a mechanism which can explain the bubble formation phenomenon. A theoretical equation has been developed on the basis of the mechanism and it is found to be applicable to a considerable extent. The proposed mechanism is for bubble formation in viscous liquids under constant flow conditions. In order to collect the necessary data for testing the validity of the proposed theoretical equation, an apparatus has been designed. It consists of a constant pressure arrangement to give an air stream under constant pressure. This is achieved by allowing water to flow into an aspirator bottle from a constant level tank. By the flow of water into the aspirator bottle the air gets compressed. Arrangements are provided to dry this air and then saturate it with the particular liquid in which the bubbles are formed. The air supply is let into an eductor tube by means of a needle valve. The capillaries or orifices used are fixed at the bottom of the eductor tube in which the liquid is filled. The bubbles formed at the capillaries or orifices are allowed to pass through a soap-film flow meter using which the bubble volume measurements are made. Six capillaries of diameters ranging from 0.0692 cm to 0.312 cm have been employed during the present investigation, and bubble frequencies up to about 200 bubbles/minute have been studied for liquids of viscosities varying from 100 to 908 centipoises. In the experiments with the air–water system with capillaries, the bubble volume decreased with increasing flow rates at the very beginning, then started increasing with flow rate. But when the same capillary was filled with glass powder and the experiments were conducted, it was seen that the bubble volume never decreased with increasing flow rate but showed an increasing trend right from 0 cm³/sec flow rate. This difference in trends in these two cases may be explained as follows: In the former case, the operation was not carried out under conditions of constant flow, whereas a constant flow condition could be ensured in the other case. From the experiments conducted with viscous liquids the following observations are made:

The gas flow rate affects the bubble volume to different extents for liquids of different viscosities. The diameter of the capillary comes into play at the initial stages of formation by providing increased circumference for the surface tension force to act on, and also during the detachment stage by providing a higher area for gas flow. The effect of surface tension of the liquid on the bubble formation exists only during the inflation of the bubble at the tip of the capillary and not during the motion and detachment of the bubble. The liquid viscosity affects the bubble formation phenomenon in a complex manner. The bubble volume increases with liquid viscosity and also the rate of increase is high for high viscous liquids. Further, with the increase in capillary diameter the influence of viscosity becomes predominant.

The various models available in the literature have been analysed, and the equations given by different investigators have been employed for calculating the bubble volume using the data collected during the present investigation. Since most of the equations are empirical in nature, the comparison is not very satisfactory. Therefore in the beginning a semi?theoretical equation has been developed in the following lines. According to this Model I, the bubble is considered to be formed in two stages, i.e., the stage of initial bubble formation and secondly the stage of detachment. During the first stage the following equation is assumed to be applicable. For the second stage the Stokes’ equation is considered to be applicable and on these grounds a final expression for the bubble volume VbV_bVb? is derived: Vb=4.270D?0.068v?1.247or0.052gv5/5S(2)V_b = 4.270D - 0.068\sqrt{v} - 1.247 \quad\text{or}\quad 0.052g\frac{v^{5/5}}{S} \tag{2}Vb?=4.270D?0.068v??1.247or0.052gSv5/5?(2) The bubble volumes have been calculated using the above equation and compared with the experimental values. The maximum deviation is about 10% and most of the calculated values fall within 5% of the experimental ones. Model II is also based on a two?step mechanism. In the first stage the bubble grows at the tip of the capillary and does not move as a free bubble. Throughout this stage the sum of the downward forces is larger than the upward buoyancy force, though the difference between the two goes on getting reduced as the bubble expansion continues. The first stage is considered to be complete when the buoyancy force equals the total downward force. After completion of the first stage, the upward forces become higher and the bubble starts moving upwards. The bubble does not move with a constant velocity but actually accelerates. This upward ascent continues until the bubble breaks off. Until this stage, the bubble continues to obtain the supply of gas from the main stream. Thus the final bubble volume VbV_bVb? is given by the equation: Vb=Vgi+Qt(5)V_b = V_{gi} + Qt \tag{5}Vb?=Vgi?+Qt(5) where VgiV_{gi}Vgi? is the force?balance bubble volume obtained from: Vgi?(?l??g)?g=2?r(34?)1/3(4)V_{gi} \, ( \rho_l - \rho_g ) \, g = \frac{2\sigma}{r} \left( \frac{3}{4\pi} \right)^{1/3} \tag{4}Vgi?(?l???g?)g=r2??(4?3?)1/3(4) To determine the QtQtQt value, the value of ttt has to be evaluated. Two criteria have been proposed to get the value of ttt. Using the first criterion, tc=C(NM)t_c = C \left(\frac{N}{M}\right)tc?=C(MN?) According to the second criterion, t=Vl(5)t = \sqrt{V_l} \tag{5}t=Vl??(5) During the present studies it was found that the first criterion of detachment is applicable to glass capillaries and the second criterion of detachment is applicable to stainless?steel orifices. Using these equations the bubble volumes have been calculated and compared with the observed ones, and the agreement has been found to be good. The effect of orifice geometry on bubble formation in viscous liquids has also been studied during the course of the present study. The different geometries of the orifices which were used are triangle, square, pentagon and hexagon. Experiments have been conducted in viscous liquids of three different viscosities. Before attempting to correlate the results, the equivalent?dimension concept based on the following was tried:

Diameter basis Perimeter basis Area basis

None of the above three could satisfactorily take into account the variation in bubble size for geometrical orifices. Two new parameters were defined as below: P?=Area of the non?circular orificeArea of the inscribed circular orificeP' = \frac{\text{Area of the non?circular orifice}}{\text{Area of the inscribed circular orifice}}P?=Area of the inscribed circular orificeArea of the non?circular orifice? ?=Bubble volume from the non?circular orificeBubble volume from the inscribed circular orifice(7)\alpha = \frac{\text{Bubble volume from the non?circular orifice}}{\text{Bubble volume from the inscribed circular orifice}} \tag{7}?=Bubble volume from the inscribed circular orificeBubble volume from the non?circular orifice?(7) Using these two parameters it was possible to evaluate an equation using which we could get the bubble volume from the non?circular orifice once we knew the inscribed circular orifice diameter and the experimental condition. The final relationship is: Vd=Vc×?V_d = V_c \times \alphaVd?=Vc?×? To obtain the bubble volume from the circular orifice the equation (3) can be used.